On the Nodal Line Conjecture

Abstract

We consider Dirichlet eigenfunctions of membrane problems. A counterexample to Payne’s nodal line conjecture is given, i.e. a domain in R2 (not simply connected) whose second eigenfunction has a nodal set disjoint from the boundary. Also a domain in R2 is given whose second eigenvalue has multiplicity three. Furthermore, some sufficient conditions are given which imply that an eigenfunction of a Dirichlet membrane problem in Rn has a zero set which hits the boundary.